Unwrapping sphere in order to fit this image
We can unwrap the sphere same as a cube, if the geometry of sphere is just subdivided and displaced cube:



Create a cube and give it these modifiers:



Proceed same as with unwrapping a cube. The texture can look like this:



Notice the bleeding over the unwrapped mesh. That is to not pollute the surface with green (or black in my example). You can also pack those faces much more efficiently if you like, to not waste any texture space: